Initial value problem for the free-boundary magnetohydrodynamics with zero magnetic boundary condition

We show local well-posedness of fluid-vacuum free-boundary magnetohydrodynamic (MHD) with both kinematic viscosity and magnetic diffusivity under the gravity force. We consider three-dimensional problem with finite depth and impose zero magnetic field condition on the free boundary and in vacuum. Sobolev–Slobodetskii space (fractional Sobolev space) is used to perform energy estimates. Main difficulty is to control strong nonlinear couplings between velocity and magnetic fields. In [D. Lee, SIAM J. Math. Anal., 49(4):2710–2789, 2017], we send both kinematic viscosity and magnetic diffusivity to zero with same speed to get ideal (inviscid) free-boundary magnetohydrodynamics using the result of this paper.

Keywords

magnetohydrodynamics, free-boundary problem

2010 Mathematics Subject Classification

35Q30, 35Q35, 76B03

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The author was partially supported by NSF grant DMS-1211806.

Received 10 April 2017

Received revised 1 October 2017

Accepted 1 October 2017

Published 30 August 2018